Factorial
The factorial is a function applied to whole numbers, defined asFactorial from Wolfram MathWorldFactorials from PurpleMath $$n! = \prod^n_{i = 1} i = n \cdot (n - 1) \cdot \ldots \cdot 4 \cdot 3 \cdot 2 \cdot 1.$$ For example, \(6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 720\). It is equal to the number of ways \(n\) distinct objects can be arranged, because there are \(n\) ways to place the first object, \(n - 1\) ways to place the second object, and so forth. The special case \(0! = 1\) has been set by definition; there is one way to arrange zero objects. Before the notation \(n!\) was invented, \(n\) was common. The function can be defined recursively as \(0! = 1\) and \(n! = n \cdot (n - 1)!\). The first few values of \(n!\) for \(n = 0, 1, 2, 3, 4, \ldots\) are 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, and 39916800. Properties The sum of the s of the factorials is \(\sum^{\infty}_{i = 0} i! = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots = 2.7182818285\ldots\), a mathematical constant better known as \(e\). In fact, \(e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\), which illustrates the important property that \(\frac{d}{dx}e^x = e^x\). Because \(n! = \Gamma (n + 1)\) (where \(\Gamma (x)\) is the ), \(n! = \int^{\infty}_0 e^{-t} \cdot t^{n + 1} dt\). This identity gives us factorials of positive real numbers (and negative non-integer real numbers), not limited to integers: *\(\left(\frac{1}{2}\right)! = \frac{\sqrt{\pi}}{2}\) *\(\left(-\frac{1}{2}\right)! = \sqrt{\pi}\) In base 10, only two non-trivial numbers are equal to the sum of the factorials of their digits: \(145 = 1! + 4! + 5!\) and \(40585 = 4! + 0! + 5! + 8! + 5!\). The number of zeroes at the end of the decimal expansion of \(n!\) is \(\sum_{k = 1} \lfloor n / 5^k\rfloor\).Factorials and Trailing Zeroes from PurpleMath For example, 10000! has 2000 + 400 + 80 + 16 + 3 = 2499 zeroes. In general, the number of zeroes at the end of the base-\(b\) expansion of \(n\)! is \(\sum_{k = 1} \lfloor n / p^k\rfloor\), where \(p\) is the largest prime factor of \(b\). Specific numbers One hundred factorial's decimal expansion is shown below. : In scientific notation, this is approximately 9.3326215443 × . It seems approximately , although larger almost 100 million times. Lawrence Hollom calls 200! faxul. One thousand factorial is about 4.0238726007 × . Variation Aalbert Torsius defines a variation on the factorial, where \(x!n = \prod^{x}_{i = 1} i!(n - 1) = 1!(n - 1) \cdot 2!(n - 1) \cdot \ldots \cdot x!(n - 1)\) and \(x!0 = x\).http://c2.com/cgi/wiki?ReallyBigNumbers \(x!n\) is pronounced "n''th level factorial of ''x." \(x!1\) is simply the ordinary factorial and \(x!2\) is Sloane and Plouffe's superfactorial \(x\$\). The special case \(x!x\) is a function known as the Torian. Pseudocode // Standard factorial function function factorial(z''): ''result := 1 for i'' '''from' 1 to z'': ''result := result * i'' '''return' result // Generalized factorial, using Lanczos approximation for gamma function g'' := 7 ''coeffs := 676.5203681218851, -1259.1392167224028, 771.32342877765313, -176.61502916214059, 12.507343278686905, -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7 function factorialReal(z''): ''ag := coeffs0 for i'' '''from' 1 to g'' + 1: ''ag := ag + coeffs[i''] / (''z + i'') ''zg := z'' + ''g + 0.5 return sqrt(2 * pi) * * * ag // Torsius' factorial extension function factorialTorsius(z'', ''x): if x'' = 0: '''return' z'' '''if' x'' = 1: '''return' factorial(z'') ''result := 1 for i'' '''from' 1 to z'': ''result := result * factorialTorsius(i'', ''x - 1) return result Sources See also ja:階乗 Category:Functions Category:Factorials